Skip to main navigation menu Skip to main content Skip to site footer

Uniqueness of a power of an entire function with its differential-difference polynomials sharing a value CM

Abstract

In this paper, we investigate the scenario when power of a transcendental entire function shares a value CM with its differential-difference polynomial. Our result extend and generalize a recent result of Adud and Chakraborty [1].

2020 Mathematics Subject Classification:

30D30, 30D20, 30D35

Keywords

value sharing, weighted sharing, shift, difference polynomial

Full text

Author Details

Soumon Roy

Nevanlinna Lab, Department of Mathematics
Ramakrishna Mission Vivekananda Centenary College
Rahara, West Bengal 700 118, India
e-mail: rsoumon@gmail.com

Sudip Saha

Department of Mathematics
Brainware University
Barasat, Kolkata, West Bengal 700 125, India
e-mail: sudipsaha814@gmail.com


References

  1. Md. Adud, B. Chakraborty. Some results related to differentialdifference counterpart of the Brück conjecture. Commun. Korean Math. Soc. 39, 1 (2024), 117–125.
  2. R. Brück. On entire functions which share one value CM with their first derivative. Results Math. 30, 1–2 (1996), 21–24.
  3. Y.-M. Chiang, S.-J. Feng. On the Nevanlinna characteristic of f(z + η) and difference equations in the complex plane. Ramanujan J. 16, 1 (2008), 105–129.
  4. Z. Chen. On the difference counterpart of Brück’s conjecture. Acta Math. Sci. Ser. B (Engl. Ed.) 34, 3 (2014), 653–659.
  5. X. Dong and K. Liu. Some results on differential-difference analogues of Brück conjecture. Math. Slovaca 67, 3 (2017), 691–700.
  6. M. Fang. Uniqueness of admissible meromorphic functions in the unit disc. Sci. China Ser. A 42, 4 (1999), 367–381.
  7. G. G. Gundersen. Meromorphic functions that share three or four values. J. London Math. Soc. 20, 3 (1979), 457–466.
  8. G. G. Gundersen. Meromorphic functions that share four values. Trans. Amer. Math. Soc. 277, 2 (1983), 545–567.
  9. G. G. Gundersen, L.-Z. Yang. Entire functions that share one value with one or two of their derivatives. J. Math. Anal. Appl. 223, 1 (1998), 88–95.
  10. R. G. Halburd, R. J. Korhonen. Difference analogue of the lemma on the logarithmic derivative with applications to difference equations. J. Math. Anal. Appl. 314, 2 (2006), 477–487.
  11. R. G. Halburd, R. J. Korhonen. Meromorphic solutions of difference equations, integrability and the discrete Painlevé equations. J. Phys. A. 40, 6 (2007), R1–R38.
  12. W. K. Hayman. Meromorphic functions. Oxford Math. Monogr. Oxford, Clarendon Press, 1964.
  13. J. Heittokangas, R. Korhonen, I. Laine, J. Rieppo, J. Zhang. Value sharing results for shifts of meromorphic functions, and sufficient conditions for periodicity. J. Math. Anal. Appl. 355, 1 (2009), 352–363.
  14. X.-H. Hua. Sharing values and a problem due to C. C. Yang. Pacific J. Math. 175, 1 (1996), 71–81.
  15. Z.-B. Huang, R.-R. Zhang. Uniqueness of the differences of meromorphic functions. Anal. Math. 44, 4 (2018), 461–473.
  16. I. Lahiri. Weighted value sharing and uniqueness of meromorphic functions. Complex Variables Theory Appl. 46, 3 (2001), 241–253.
  17. E. Mues, N. Steinmetz. Meromorphe Funktionen die mit ihrer Ableitung Werte teilen. Manuscripta Math. 29, 2–4 (1979), 195–206.
  18. L. A. Rubel, C. C. Yang. Values shared by an entire function and its derivative. In: Complex analysis (Eds J. D. Buckholtz, T. J. Suffridge) (Proc. Conf., Univ. Kentucky, Lexington, Ky., 1976), 101–103. Lecture Notes
  19. in Math., vol. 599. Berlin, Springer, 1977, https://doi.org/10.1007/BFb0096830.
  20. C. C. Yang, X. H. Hua. Uniqueness and value sharing of meromorphic functions. Ann. Acad. Sci. Fenn. Math. 22, 2 (1997), 395–406.
  21. H. X. Yi. Meromorphic functions that share one or two values. Complex Variable Theory Appl. 28, 1 (1995), 1–11.
  22. Q. C. Zhang. Meromorphic function that shares one small function with its derivative. JIPAM. J. Inequal. Pure Appl. Math. 6, 4 (2005), Article 116, 13 pp.